ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 311 



have a continuous series from the smallest air-angle to an oil-angle 

 of 180° (the air-angle of 180° being by no means a maximum), 

 it is obvious that the only true and scientific notation for the com- 

 parison of apertures must necessarily be progressive also, and that 

 no justification can be found, even as a matter of convenience, for 

 the adoption of one which first advances from 1 to 180, and then 

 instead of going forward goes bach to 96 and for a second time to 

 180, then back once more to 127, and on for the third time to 180. 

 If nothing more could be said against such a notation than its want 

 of scientific precision, it might be allowed to pass with only an ex- 

 pression of surprise that any one could desire to retain it, but the 

 mischief of the notation goes beyond any question of taste merely, in 

 that it misleads the microscopist into supposing that the second and 

 third 180°, being the same figures, represent essentially the same 

 aperture, and so obscures the one important and practical point in 

 Connection with aperture. 



Whilst the fact of the progressive increase in the diameter of 

 the emergent beam, i. e. in the number of rays emitted by the objective 

 at its back surface, might have been supposed to be abundant proof of 

 itself that there must have been a similar increase in the number of 

 rays admitted from the object by the front surface, — sufficiently dispos- 

 ing therefore of the principle on which angular aperture is based, and 

 necessarily leading also to the recognition of the proper notation for 

 aperture, — it is rarely that the angular aperturist is content with this 

 mode of dealing with the matter. He considers that " his points " 

 have not been directly met, which he more than suspects is due, not 

 to the fact that they have no basis, but because they are incon- 

 veniently sound. 



Before, therefore, passing to the determination of the true aperture 

 notation, it will be desirable to show that 180° angular aperture in 

 air does not in fact represent any natural limit or maximum, either 

 (1) pliotometrically, or in regard to the number of rays ; or (2) as a 

 question of resolution ; or (3) by virtue of what is known as " angular 

 grip" The fact also that the use of the angular expression is mis- 

 leading and erroneous even in the case of the same medium, may 

 conveniently be shown at the same time. 



(5) The Photometrical Test. — Supposed Identity of the Hemi- 

 spheres in di£Ferent Media. — The point which the angular aperturist 

 almost invariably takes up first is the photometrical one, as he con- 

 siders that to fui-nish the most unassailable proof that 180° in air 

 represents a " whole " which may be equalled but never exceeded. 



With the same fixed illumination, 180° in oil cannot, he supposes, 

 represent anything in excess of 180° in air as regards quantity of 

 light, and pencils of any given angular extension (say 82°) in oil are 

 necessarily only equal, therefore, in that respect to the same pencils 

 in air. As there can be no more than the hemisphere in angular 

 measurement, and as he assumes radiation to be the same in all 

 media, it is self-evident, he thinks, that with the hemisphere in air we 

 have a loliole of light, beyond which there can be nothing. This 

 whole can approximately be taken up by a dry lens, and being the 



